COUNTIF can't count non-blank formula results?

Question

I am looking for a criterion to use in =countif(range_of_formulas, [criterion]) that will identify non-blank results of formulas. I've found quite a few of those, like "<>" , "<>"&"" , "<>""" , ">""" , "*" and more (e.g. on SO here and here). But each criterion seems to work only in some situations and not others, and they all fail in the following situation:

columns A and B have my original data. Column D presents results of the formulas you see in column C. Column F shows results of countif using different criteria (as shown in column E) to count the non-blank results in column D. None of them arrive at the right count - 5.

In my research (and in answers to this question, before I edited to narrow it down) I've seen many workarounds that would get the right count. But what I am asking is if there is a criterion that would work in countif, or if countif cannot reliably count non blank formula results?
(or maybe someone can say under which conditions it can or can't be used).

Answer 1

This can be achieved with a simple change on the first forumla to force the output to be a string value (Should you need to perform a calculation with the numerical results, multiply them by 1 to convert back to number when they are being used)

I have simply joined a blank string to the end of the result on the first formula and wrapped it in brackets to keep excel happy:

=(IF(AND(B1=1,ISBLANK(A1)=FALSE),A1,"")&"")

Now you can use the wildcards for any character (?) and any length of string (*) together as your criteria to achieve your desired total of 5:

=COUNTIF($D$1:$D$8,"?*")